Lecture 12:
Sensitivity Examples
(Shadow Price Interpreted)
AGEC 352
Spring 2012 – February 29
R. Keeney
Shadow Price signs


Signs on shadow prices differ whether
the inequality constraint is ≤ or ≥.
They also differ for maximization and
minimization problems.
Maximization Minimization
≤
Positive
Negative
≥
Negative
Positive
Less than (<=) case
A boundary that is <= (upper bound)
 We use +1 definition of shadow price

◦ The +1 will always ‘relax’ the upper bound

A decision maker facing a less restrictive
choice set
◦ Can be better off (binding constraint)
◦ Can be unaffected (slack constraint)

Better off depends on max vs. min
Great than (>=) case
A boundary that is >= (lower bound)
 We use +1 definition of shadow price

◦ The +1 will always ‘tighten’ a lower bound

A decision maker facing a more restrictive
choice set
◦ Can be worse off (binding constraint)
◦ Can be unaffected (slack constraint)

Better off depends on max vs. min
Example (Upper/Max)

Upper bound
◦ Maximization
◦ Land available to plant
 Shadow price = the change in returns generated by
a +1 to the land constraint
 Shadow price = Maximum rent that can be paid
 Use extra profits from additional resources to acquire the
resource
Example (Upper/Min)
Upper bound
 Minimization

◦ Fertilizer mix phosphate limit
◦ Shadow price = the change in costs from a 1
unit increase in the phos limit
◦ Shadow price = discount the mixer could
offer to the buyer to expand the phos limit
 Pass some of cost savings to buyer
Example (Lower/Max)
Lower bound
 Maximization

◦ Every 10 acres of corn planted requires 1
acre left fallow (set aside)
 Shadow price = change in profits from increasing
set-aside by 1
 Shadow price = payment farmer must receive to
participate
Example (Lower/Min)
Lower bound
 Minimization

◦ Calcium requirement in a daily diet
 Shadow price = change in cost of requiring an extra
unit of calcium
 Shadow price = maximum price that can be paid
per unit of non-food calcium supplement
Lab Assignment Problem

4 Fertilizers (see lab 5 for fertilizer info)
◦ Different compositions of nitrogen, potash,
and phosphate
◦ Meet an order (at minimum cost) by mixing
the four fertilizers that has:




Exactly 1000 units of fertilizer
At least 20% (by weight) nitrogen
At least 30% (by weight) potash
At most 8% (by weight) phosphate
Shadow Prices in Fert. Problem
Fertilizer
Component
LHS
RHS
Shadow
Price
Nitrogen
201.3
>= 200
0.00
Potash
300.0
>= 300
10.00
Phosphate
80.0
<= 80
-14.00
Total Weight
1000
=1000
11.70
Interpretation of Potash

Potash constraint
 Required to have a minimum amount of potash
in the fertilizer mix
 Increasing the RHS of the potash constraint
makes the problem more restrictive, higher
percentage of potash required
 Shadow price is positive because costs will
increase with the increase of RHS
 Interpret this as the amount we would be willing
to pay to avoid having the RHS increase
 Also, the discount we could offer for a mix that
had 0.1% less potash content
Interpretation Phosphate

Phosphate constraint
 Upper limit on the phosphate content
 Increasing the RHS of the phosphate
constraint makes the problem less restrictive,
higher percentage of phosphate allowed
 Shadow price is negative because costs will
decrease with the increase of RHS
 Interpret this as the amount we would be
willing to pay to relax the RHS by one unit
 Also, the markup we should charge if
someone required 0.1% less phosphate in their
fertilizer mix
Interpretation in general

Always should be in context of the
problem
◦ Signs are actually trivial if you understand
the problem (better off/worse off)
◦ Does an increase in the RHS improve or
worsen the objective?
 If it improves, then we know the willingness to
pay for increasing the RHS
 If it worsens, then we know the willingness to pay
to avoid having the RHS increase
Advanced Analysis:
Which constraint is the most costly?

Recall the cereal problem from lecture
◦ Two cereals mixed to meet minimum
requirements on thiamine, niacin, and calcium
Nutritional
Requirement
LHS
RHS
Shadow Price
Thiamine
1
>= 1
14.44
Niacin
5
>= 5
2.36
Calories
722.2
>= 500
0.00
Rather than comparing units, we
want to compare % of RHS
1 mg of thiamine and 1 mg of niacin are
not directly comparable
 % increases in the RHS of constraints
are however

Nutritional
Requirement
Thiamine
Niacin
Calories
RHS
1
5
500
1%
increase
0.01
Shadow
Price
14.44
SP * 1%
increase
0.14
0.05
5
2.36
0
0.12
0.00
Ranking the constraints

Thiamine was the most costly constraint
to meet
 We would have judged this the same just
comparing shadow prices, but that could be
misleading

Similar to elasticity interpretations
 Elasticity of demand for food versus cars

Requires that you understand the
problem and interpretation to make the
comparisons
Fertilizer Problem
Fert Component
N
K
P
total

RHS
200
300
80
1000
Shadow Price
0
10
-14
11.7
1 pct
2
3
0.8
10
Value of 1Pct Increase
0
30
-11.2
117
Consider
◦ Is total comparable to others?
◦ How to deal with positive vs negative shadow
prices?
 Compare relaxations of constraints…
Common percentage and direction
(of objective variable)
Fert Component
N
K
P
total

RHS
200
300
80
1000
Shadow Price
0
10
-14
11.7
1 pct
2
3
0.8
10
Cost saving, 1% change in K
◦ Total cost reduces $30.00

Cost saving, 1% change in P
◦ Total cost reduces by $11.20
Value of 1Pct Increase
0
30
-11.2
117
Planting Problem
Land LHS
Rowcrop Land LHS
Wht Allot LHS
Jan-Apr Labor LHS
May-Aug Labor LHS
Sep-Dec Labor LHS

500
400
120
1600
2000
1600
1% S Price
SP * 1%
5
13.75
68.75
4
0
0
1.2
12
14.4
16
6.25
100
20
0
0
16
0
0
Shadow price for land is 2X labor
◦ 1 unit of land is usually worth more than a
unit of labor

Compare them as 1% increase in our
resource base (labor > land > allot)